Digital download
I have achieved a first-class in Math by taking these notes. First-year uni math notes in SMA for those who struggle through the course or for uni applications interview.
Hope can help you pass the exam
Cheers
i. Definition scalar :
A scalar is a number which describes some physical
-
quentieg
um
g Temperature T
e.
Kelvin
.
,
A
,
scalar comes with
it second
Time
-
,
→
a set
of units
Density P
kgm
-
, ,
A scalar be constant also
but
depend
2.
can
, can on another quantity .
THI denote that
temperature T depends time t
g
e. -
on
Vectors :
i. Definition vectors : are quarries with both direction and a ( scalar )
magnitude .
"
e.
g
.
velocity , I , ms
-2
Each vector also
•
Acceleration , a , ms
-
has unit associated
Force I N
'
, , to it
"
-
The force I exerted by an electric field , NC
"
The
force I exerted
by a
magnetic field , T
Notation
U→
2.
usually
:
We underline I arrow
or
place top to
.
an on
mum
mum
denote the u is a vector
•
Bald letter
for print
} .
As with scalars ,
a vector can
depend on some other
gmonties
e.
g E th
Directions talk about directions need frame reference
4. : To we a
of
0
-
Origin
5- position vector :
think of
We can a vector I as
representing the
displacement of a
point A
in
origin space
relative
>
to the
.
.
A
a
I
.
0A.
a- is
position .
vector
of A y
=
,
the
c.
Magnitude
:
6.
•
C
magnitude
= IB71 =
III
g
e.
7
I
8
B
7- Unit Vector :
by £
we call vector of 1 unit vector and denote
length it
we
any a
me
8. A vector can
represent the
displacement between two distinct pairs of
points
, •
A .
C
→ both vectors have the same
7
>
and
°
&
•
direction
magnitude , so
they are the same vector
B
i. II = BE
f. Observations :
In words the direction and under
• other .
magnitude of a vector are invariant
translation .
scalar
magnitude of vector is invariant under
•
The a rotations .
1. 2 . Vector
Algebra
Vector addition and the zero vector :
'
from 0 travel I 0A A
starting the
origin along vector point
=
i. , some to a .
Then travel
along vector to ,
and reach the
point C
A.
±
a-
7
.
(
Y c
a.
-
have
The
point
C must
already a
position vector
→
I = 0C
A
-
.
. I + I =
I
b-
I s
addition
z .
Properties of vector :
+
<
① a- + I = I + I [ vector addition is commutative ] d U
Ta
-
±
② 11th ) + of =
a- + It + d- I [ vector addition is associative ]
( III a)
The
position vector of the
origin ,
0 is
zeirovec-or.fm =
considered don't
It is as a vector .
Only vector have direction
③ a- + A =
It I = I [ the zero vector is the
additive
identity I
otit At = e
If OTI =
I ,
then AT = -
I -
I is called the additive inverse of I
4 a- + 1- E) E) E tbf )
E- I
=
1- + I =
I =
a- +
Scalar
Multiplication →
:
• let a- = 0A and I =
.
Also I =
tea
.
.
.
I has the same direction as I ,
but
only ¥ the
length
i. In
general , for any
real number R . we combine it a vector a to
get a new vector
I =
the
•
when I > a , I has the same direction as I
when I I has the direction as I
20
opposite
- .
e.
g. 7
£
> -
f
Ee L
• when I = a , I = a
Two vectors are
parallel if one is of the form DX other vector
( or anti -
parallel when i ca )
, of scalar multiplication
Properties
:
> .
① I X I = I
② For scalar X
any ,
he + it = A
IITII
③ For scalar and
any X ll .
XI -1
ME = 1 Atm ) I
④ For scalar and
any X µ ,
Rima ) = KM ) I
1. 3 Vector and Coordinate Bases
space
Vector :
space
addtion
1-
Definition Vector space , V :
any hmm
collection of objects equipped with the two
operations 1 Vector
and scalar multiplication ,
satisfying the eight properties outlined I
vectors
z . Vector :
The
objects
um
themself are then
referred to as
space III space )
will restrict attention to the real vector ( two dimensional Euclidean and
3 We our
-
.
.
IRI ( three - dimensional Euclidean
space
) in this course .
.
Scalars are the real number IR
m u m
I A frame of reference 1
origin axis axis and axis
× y Z
together with
:
4 an - -
.
a
-
.
, ,
unit
length I
to ,o a)
• The origin ,
&
PIX y ZI Cartesian Coordinates
-
, '
position vector of P : IF = I =
( If )
of Ipl =] x 't
y -122
'
magnitude
• a vector :
•
vector Addition : If I
=
( £! ) and I =
(If;) , then :
" ± -1%1+11*1=1%11%1
scalar
multiplication for scalar R in IR
•
:
any
NII '
Questions :
suppose that a. =
1¥ ) and b- =/! ) .
Then :
al ath
att -
-
1%1+111=111
b-I -3A
→a =
-4%1 =
II )
4 lol
lol =/ o't 5+1-412 = 5
is .
We can write
any
vector p= I '¥ ) in IR3 in a
unique way
as the linear combination :
(E) =\ ! ) +
yl ! ) +
ZI ! ) = +
yitzk
• The vector I , I , I are called the
¥ivectors
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller BYRZ. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $5.70. You're not tied to anything after your purchase.
